2010 AMC 12B Problems/Problem 8
Contents
Problem 8
Every high school in the city of Euclid sent a team of students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed
th and
th, respectively. How many schools are in the city?
Solution
There are schools. This means that there are
people. Because no one's score was the same as another person's score, that means that there could only have been
median score. This implies that
is an odd number.
cannot be less than
, because there wouldn't be a
th place if there were.
cannot be greater than
either, because that would tie Andrea and Beth or Andrea's place would be worse than Beth's. Thus, the only possible answer is
.
Solution 2
Let be Andrea's score. We know that she was the highest on our team, so
.
Since is the median, there are
to the left and right of the median, so the total number of people is
and the number of schools is
. This implies that
.
Also, since is the rank of the last-place person, and one of Andrea's teammates already got 64th place,
.
Putting it all together: and
, so clearly
, and the number of schools as we got before is
.
~adihaya
See also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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